Common Core Algebra II.Unit 2.Lesson 4.The Domain and Range of a Function
Algebra 2
Learning Common Core Algebra II.Unit 2.Lesson 4.The Domain and Range of a Function by EmathInstructions
Hello and welcome to another common core algebra two lessons. By E math instruction. My name is Kirk Weiler and today we're going to be doing unit two lesson four on the domain and range of a function. Now domain and range are two concepts that you saw back in common core algebra one. But they can be kind of tricky. So we're going to start off today by reviewing their basic definitions. Let's begin. The domain and range of a function. Now keep in mind, that a function is simply a rule that changes input values into output values.
The domain is the set of all allowable inputs IU ones that result in an output. Allowable, right? That sounds like a strange word. Well, what does that mean? Let's say that let's say that N was the number of texts. That we send. And let's say that we signed up for some kind of plan that charged us 25 cents per text. So this would be the formula. The function that would convert the number of texts that we sent into how much we were charged. Well, certainly something like ten would be part of the domain. I mean, great. Yeah, I can put ten texts in and figure out how much it's going to cost me. On the other hand, something like negative 5, that wouldn't be part of the domain because we can't have a negative number of texts. Likewise, we couldn't even have something like 7.2, because that doesn't make sense, right? Now ten negative 5 and 7.2 are all happily. Put into this formula. But the plain fact is the only ones that are allowable are things like ten. So the domains kind of tricky.
On the other hand, the range is pretty simple. It's simply the set of all outputs of the function. So once we have our domain, all the allowable inputs, that kind of dictates all the allowable outputs. Remember that independent and dependent idea? Okay, well let's clear this out and get into some domain and range problems. All right, exercise one, consider the function that has inputs as inputs the months of the year, January, February, march, et cetera and its outputs are the number of days, the number of days in each month. In this case, the number of days is a function of the month of the year, you know, in other words, if you say march, I'm going to tell you 31, right? If march is the input, 31 is the output. It says assume that this function is restricted to non leap years. That's actually quite important because if not, then February would have two outputs, both 28 and 29. And as we know, that's not allowed with functions.
Anyway, let array says right in roster form, we'll talk about what a roster is in a bit. Right in roster form the set that represents this function's domain. While the domain are all the inputs, that's January. February. March. Et cetera. I'm not going to write them all out. You get the picture. There's 12 of them. And December. A roster is simply a list. Now, very often, we can not actually put domains and ranges in roster forms, because there's simply too many elements in the domain and range, perhaps there's an infinite number of them, perhaps we can't order them. Here, there's 12 of them. So we can write them out. Likewise, letter B says right in roster form. It should say roster form. The set that represents this function's range. Well, this might be a little bit trickier, but if we restrict ourselves to non leap years, then a month can either have 28 days if it's February. Or 30 or 31 days. That's the range of our function. All right? So it's pretty easy. The domain is all the allowable inputs, the range is all the outputs.
By the way, think about the fact that there's many things that just aren't allowable for the domain. So for instance, you wouldn't allow the domain to include the state of Illinois. Because Illinois is not a month. Therefore, it's simply not allowable for this function. All right? So it's kind of cool. Anyhow, let's do some more work with this, pause the video now if you need to write any of this down, and then we'll clear out the text. Okay, here we go. Exercise two, it says state the range of the function F of N equals two N plus one if its domain is the set one comma three comma 5. Show the domain and range in the mapping diagram below. So very often we show functions by showing what's known as a mapping diagram. And in mapping diagrams often the domain and range. Are shown as ovals containing the inputs and the output. So I'm going to put my domain of one three 5 in there. Keep in mind that the rule is take the input and multiply it by two. And then add one to it. So if I take one as an input, multiply it by two, I get two, add one to it, and I get three.
So an input of one gets converted to an output of three. On the other hand, if I take three and I multiply it by two, I get 6. And I add one, and I get 7. And one more time, two times 5 is ten, plus one is 11. So my range. Is three comma 7. 11. In roster form. All right, so that was that was easy. Let's clear this out and go on to ones that are a little bit more complicated. Oh, a graph. The function Y equals G of X is completely defined by the graph shown below. So there is no more than just what we see. There's no little arrows, the graph doesn't extend forever. Answer the following questions based on this graph. Letter ace has determined the minimum and maximum X values represented on this graph. Well, the minimum X value is going to be all the way over on the left side. So I think I'll do X min equals what's that negative three. Let's see, my X max is going to be over here. One, two, three, four, 5, 6. So X max is 6. A little bit harder to determine the minimum and maximum Y values. Let's see, here's my minimum Y value. And it's going to be at one, two, three, four, 5, negative 5. And let's do the maximum. Let's see, that's going to be right here. And that's going to be three.
Now letter C says state the domain and range of this function using set builder notation. Let me show you how that looks. Well, there's no way to do this in roster form now because there's too many X and Y values. But we can say what the domain is all values of X such that X is greater than or equal to negative three, while being less than or equal to 6. And the range likewise is all values of Y, such that negative 5 is less than or equal to Y is less than or equal to three. That's what's called set builder notation. We're also going to use something called interval notation, but we'll get to that in just a little bit. Okay? So write down what you need to pause the video and then we'll clear this out. All right. Exercise number four. Oh, here comes interval notation. The function F of X equals X squared minus two X minus one is graphed on the grid below. Which of the following represent its domain and range written in interval notation? All right. Well, take a look at the fact that there's arrows on that. What that means is that the graph just keeps going on and on and on, heading to the left, heading to the right forever. Right? As well, notice that the minimum Y value one, two, three, four, is negative four, and then it just goes up forever. So our correct answer ends up being choice number three. All right? And it's choice number three, the domain negative infinity deposit infinity means that any X value will be able to put into this function. The range, on the other hand, starts at negative four, remember the brackets mean bracket means includes.
So it includes the negative four, and then it just goes up forever. So we don't end it infinity. By the way, notice how the infinity has parentheses on it. So square bracket means includes that bracket means not included. Okay. So I'm going to clear this out. And let's take a look at one with a table. The function F of X equals two X plus one divided by X minus four has outputs given by the following calculator table. So if I if I put this into my table and I looked at it, you know, granted, I'd see X maybe Y one, but whatever. So here are outputs. Now, letter a just says evaluate F of one and F of 6. Again, this is awesome. I love tables for evaluating functions. Here's my input. And my output is negative one. When my input is 6, my output is 6.5. Now, whoops. Sorry about that. Now letter B says, why does the calculator give an error at X equals four? Why does that happen? Well, let's take a look. Let's try to figure out what F of four is. While the rule says two times four plus one divided by four minus four. That's going to be 8 plus one is 9 in the top and of course four divided by four is zero. Now it may seem that 9 divided by zero would be zero, but in fact, what you've seen a little bit, you haven't had a lot of exposure to this, but is that you can not divide by zero. So that means X equals four is not in the domain.
Not in the domain. We can't come up with an output. That's why the calculator is giving us an error. Now, let her see says, are there any values except X equals four that are not in the domain of F explain? Oh no, that's okay. No. No? All other values. Are fine. But we have to worry about division. Okay, so watch out. Watch out for division. By zero. Yeah, just can't do it. Never can divide by zero. So I'm going to clear this out, pause the video again if you need to. All right, here we go. Sorry about that little jog in the screen. I hit my computer in it. It made it move. All right, let's take a look at one involving a square root. It says, which of the following values of X would not. Be in the domain of this function, explain your answer. Well, why in the world would we not be able to put a value of X in there? Well, let's try X equals zero for a second. If I put zero in here, hit the square root of four. Well, let's see equal to two. I was able to get an output, right? But one of these values of X won't work. It will not allow me to have an output. Think about which one it is.
All right. What I'm hoping is that you all concluded that this X equals negative 8 is not in the domain. And if we try it, if we try to see why that is, well, we eventually get that our output must be the square root of negative four. But yet again, and we've seen this a couple times already in this course, we can not take square roots square roots. Of negatives. All right, so that's not in the domain. All right, I'm going to clear this out. Let's keep going. So today, we looked at the two very, very important sets. That functions are concerned with. The domain of the function and the range of the function. In a higher level math, the sets are just as important as the functions themselves. In fact, they're probably more important. Because the plain fact is what a function does is it takes a domain and it converts it into a range. That sort of what it does. All right? So knowing the terminology and also knowing why division by zero and taking square roots of negatives, somewhat, if you will, eliminate values from being in the domain. All right, more on that later. I'd like to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler. And until next time, keep thinking and keep solving problems.